How to approach a summation problem with alternating terms?

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SUMMARY

This discussion focuses on solving summation problems involving alternating terms, specifically the sequence of odd numbers. The user is guided to calculate partial sums, denoted as Sn, for the series where S1=1, S2=4, and S3=9. The approach suggests making a conjecture based on observed patterns and then applying mathematical induction to prove the conjecture. The sequence is represented as 1 + 3 + 5 + ... + (2n-1), indicating a clear method for simplification and analysis.

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  • Understanding of arithmetic series and summation notation
  • Familiarity with mathematical induction
  • Basic knowledge of odd and even number properties
  • Ability to manipulate algebraic expressions
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  • Study the properties of arithmetic series and their sums
  • Learn about mathematical induction and its applications in proofs
  • Explore simplification techniques for series involving alternating terms
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Students in mathematics, educators teaching series and sequences, and anyone interested in enhancing their problem-solving skills in summation techniques.

mendem03
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I want to know how to start this problem, but don't want the answer.
If someone can just help me get started or give me a tip on how to approach these kinds of questions
 

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Well, just try to see what happens when you add the terms up to a certain term an in your sequence, to get Sn:

S1=1

S2=1+3=4

S3=1+3+5=9



And make a guess, then use induction or some other formula.

What is your guess for Sn=a1+a2+...+an ?
 
1+3+5+...+(2n-5)+(2n-3)+(2n-1) =
= 1+(2n-1) +3+(2n-3) +5+(2n-5)+...
Do you see a simplification ?
 

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